ch 14 summary - Summary Sampling (Chapter 14) 1. Say that a...

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Summary – Sampling (Chapter 14) 1. Say that a CT signal x c t ( ) is input to an ideal sampler with sample period T S . The sampler output is the DT signal x n [ ] = x c nT S ( ) , n = 0, ± 1, ± 2, (we call x n [ ] a sampled version of x c t ( ) , with sample period T S ). Then if X F ( ) is the DTFT of x n [ ] and X c f ( ) is the CTFT of x c t ( ) , we have the frequency domain relation 1) X F ( ) = 1 T S X c F n T S Λ Ν Μ Ξ Π Ο k = −∞ , F 1 2 If x c t ( ) is bandlimited to W Hz (that is, if X c f ( ) = 0 for f > W ) where 2 W < 1 T S then equation 1) reduces to 2) X F ( ) = 1 T S X c F T S Λ Ν Μ Ξ Π Ο , F 1 2 . (The value 2 W is called the Nyquist rate for x c t ( ) , and f S = 1 T S is called the sample rate . So this says: if the sample rate is greater than the Nyquist rate, then the DTFT of the sampled signal x n [ ] is just a scaled version of the CTFT of x c t ( ) , with the frequency scaling f F T S for F 1 2 , f 1 2 T S .) 2. Now suppose that a DT signal x n [ ] is input to an ideal interpolator with sample period T S . This reconstructs a CT signal x r t ( ) from x n [ ] using the equation 3) x r t ( ) = x n [ ] n = −∞ sinc t nT S T S Λ Ν Μ Ξ Π Ο . This leads to the following frequency domain relation (where X r f ( ) is the CTFT of x r t ( ) and X F ( ) is the DTFT of x n [ ] ): 4) X r f ( ) = T S X f T S ( ) , f 1 2 T S X r f ( ) = f > 1 2 T S . (Note that this says that 1 2 T S Hz is the maximum possible CT frequency in a signal that is reconstructed from a DT signal using an ideal interpolator.) If x n [ ] = x c nT S ( ) , n = ± ± , and if x c t ( ) is bandlimited to W Hz where 2 W < 1 T S , then (by using equation 2) ) equation 4) reduces to 5) X r f ( ) = X c f ( ) x r t ( ) = x c t ( ) .
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3. Equations 2) and 5) can be summarized as this: If x n [ ] = x c nT S ( ) , n = 0, ± 1, ± 2, , and if x c t ( ) is bandlimited to W Hz where 2 W < 1 T S , then x c t ( ) can be reconstructed from x n [ ]
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This note was uploaded on 10/27/2011 for the course ECE 313 taught by Professor Muschinski during the Fall '08 term at UMass (Amherst).

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ch 14 summary - Summary Sampling (Chapter 14) 1. Say that a...

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